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Register a new userOn relatively compact sets in quasi-Banach function spaces
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This paper is devoted to the study of the relatively compact sets in Quasi-Banach function spaces, providing an important improvement of the known results. As an application, we take the final step in establishing a relative compactness criteria for function spaces with any weight without any assumption.
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In this paper we develop a stochastic integration theory for processes with values in a quasi-Banach space. The integrator is a cylindrical Brownian motion. The main results give sufficient conditions for stochastic integrability. They are natural extensions of known results in the Banach space setting. We apply our main results to the stochastic heat equation where the forcing terms are assumed to have Besov regularity in the space variable with integrability exponent $pin (0,1]$. The latter is natural to consider for its potential application to adaptive wavelet methods for stochastic partial differential equations.
In this note, we show that the Hausdorff operator $H_{Phi}$ is unbounded on a large family of Quasi-Banach spaces, unless $H_{Phi}$ is a zero operator.
We develop a functional framework suitable for the treatment of partial differential equations and variational problems posed on evolving families of Banach spaces. We propose a definition for the weak time derivative which does not rely on the availability of an inner product or Hilbertian structure and explore conditions under which the spaces of weakly differentiable functions (with values in an evolving Banach space) relate to the classical Sobolev--Bochner spaces. An Aubin--Lions compactness result in this setting is also proved. We then analyse several concrete examples of function spaces over time-evolving spatial domains and hypersurfaces for which we explicitly provide the definition of the time derivative and verify isomorphism properties with the aforementioned Sobolev--Bochner spaces. We conclude with the formulation and proof of well posedness for a class of nonlinear monotone problems on an abstract evolving space (generalising in particular the evolutionary $p$-Laplace equation on a moving domain or surface) and identify some additional evolutionary problems that can be appropriately formulated with the abstract setting developed in this work.
The purpose of this note is to discuss how various Sobolev spaces defined on multiple cones behave with respect to density of smooth functions, interpolation and extension/restriction to/from $RR^n$. The analysis interestingly combines use of Poincare inequalities and of some Hardy type inequalities.
Let $X$ be an infinite dimensional uniformly smooth Banach space. We prove that $X$ contains an infinite equilateral set. That is, there exists a constant $lambda>0$ and an infinite sequence $(x_i)_{i=1}^inftysubset X$ such that $|x_i-x_j|=lambda$ for all $i eq j$.
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